Efficient measurement of point-to-set correlations and overlap fluctuations in glass-forming liquids Ludovic Berthier,1 Patrick Charbonneau,2, 3 and Sho Yaida2, a) 1)
arXiv:1510.06320v1 [cond-mat.stat-mech] 21 Oct 2015
Laboratoire Charles Coulomb, UMR 5221 CNRS and Universit´e de Montpellier, Montpellier, France 2) Department of Chemistry, Duke University, Durham, North Carolina 27708, USA 3) Department of Physics, Duke University, Durham, North Carolina 27708, USA
Cavity point-to-set correlations are a real-space tool to detect the emergence of a rough free-energy landscape accompanying the dynamical slowdown in glass-forming liquids. Measuring these correlations in model glass-forming liquids remains, however, a major computational challenge. Here, we develop a general paralleltempering method that provides order-of-magnitude improvement for sampling and equilibrating configurations within cavities. We apply this improved scheme to the canonical Kob-Andersen binary Lennard-Jones model for temperatures down to the mode-coupling crossover. Most significant improvements are noted for small cavities, which have thus far been the most difficult to study. This methodological advance enables us to study a broader range of physical observables associated with thermodynamic fluctuations. We measure the probability distribution of overlap fluctuations in cavities, which features a non-trivial temperature evolution. The corresponding overlap susceptibility is found to provide a robust quantitative estimate of the point-to-set length scale without fitting, which further reduces the computational effort. By resolving spatial fluctuations of the overlap in the cavity, we also obtain quantitative information about the geometry of overlap fluctuations, and we discuss the interplay between wall and cavity point-to-set length scales, as well as the role of surface fluctuations. PACS numbers: 64.70.Q-, 05.10.-a, 05.20.Jj I.
INTRODUCTION
A well-known difficulty in understanding the increasing sluggishness of glass-forming liquids upon lowering temperature T is that no obvious change to the static structure of the liquid accompanies it1 . In stark contrast to the critical slowing down observed near a standard critical point, static correlation functions of glass formers barely budge while the structural relaxation timescale changes by more than 15 orders of magnitude. The glassy slowdown has thus instead been attributed to a different type of criticality, one at which the free-energy landscape becomes very rugged, leading to the emergence of many metastable states separated by growing free-energy barriers2–4 . In order to capture this ruggedness, the concept of a point-to-set (PTS) correlation was introduced about a decade ago5,6 . PTS correlations generalize multi-point correlations to their infinite-point limit. Various PTS geometries have since been considered, including cavity7–11 , random pinning12–18 , and walls19,20 , and despite subtle differences between them21,22 , the overarching observation remains. For glass formers the PTS correlation length seems to grow more rapidly than lengths extracted from two-point correlation functions8,10,12,20,21 and is thus an important aspect of glassy physics. However, extracting PTS correlations from numerical simulations has thus far been limited by the computational
a) Electronic
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challenge of their measurement21,22 , and in experiments the challenge is even greater. In order to better understand the nature of this challenge, let us consider the case of PTS correlation within the cavity geometry. First consider a liquid at equilibrium, and specify a finite cavity, say a sphere of radius R, and pin everything outside of it – thus fixing a set of particles. Then allow the particles inside the cavity to explore phase space under the constraints exerted by the set of pinned particles, now acting as an effective quenched disorder. Particles inside the cavity thus explore the local free-energy landscape. A standard way of characterizing these local landscapes, first devised in the study of spin-glasses, is to consider the statistics of the overlap, q (r), between two equilibrium configurations at a point r inside the cavity. We could also define the PTS correlator as the overlap between the original configuration and a re-equilibrated configuration, rather than as between two statistically independent re-equilibrated configurations. In this paper we use the former in checking for a good equilibration, and the latter for all the evaluations of PTS observables: correlations, susceptibilities, and probability distribution functions. The major computational problem is that of properly sampling the equilibrium fluctuations of the confined fluid. The materials we consider are characterized by very slow dynamics in the bulk and confining them with amorphous boundaries makes the natural dynamics of these systems even slower20–23 . Increasing the confinement of a system using amorphous boundaries reduces the number of accessible states from an exponentially large number (in the number of
2 particles) in large cavities, to a sub-exponential one in small cavities5 . This localization in configuration space should be reflected in the evolution of the overlap between accessible configurations which we expect to grow as the volume of accessible phase space shrinks. In other words, by increasing the confinement one hopes to provoke an ‘entropy crisis’ qualitatively similar to the one which could be happening at the Kauzmann temperature, TK , in the bulk liquid24 . By working at T > TK in a finite cavity, this putative phase transition necessarily becomes a crossover, whose location defines the PTS correlation length, ξPTS (T ). In that sense cavity PTS measurements are very close in spirit to other ways of constraining the available phase space of the system that are being actively investigated, such as random pinning13–18 or coupling to a reference configuration25–30 . In all of these approaches, one is interested in understanding the emergence of metastable states in the physicallyrelevant temperature regime much above the putative Kauzmann transition, whose existence/absence is therefore completely immaterial. Previous work has only partially validated this image. Although solid evidence for a growing PTS correlation length has now been obtained in various systems8–12 , several questions remain unexplored regarding the nature of the crossover between high and low overlap regimes, the associated fluctuations, and the connection to dynamical relaxation in the bulk. A major obstacle encountered in previous studies is that the behavior of the cavities of sizes R < ∼ ξPTS has thus far been computationally hard to access due to the high free-energy barriers that separate minima in the local landscape. This computational difficulty should not come as a surprise, as the PTS crossover length corresponds to inducing the analog of an equilibrium (or ‘ideal’) glass transition in a system containing a finite number of particles5 . Recent studies of constrained systems have shown that even for a small number of particles, such an analysis indeed requires very large computer resources17,18,31 . In this paper we develop a generic computational method to more quickly and reliably sample these thermodynamic fluctuations. As a proof of principle, we study the Kob-Andersen binary Lennard-Jones model32,33 , which is a classical glass-forming liquid. Yet our approach is sufficiently generic to be applied to other glassy systems, including hard spheres, with only minor tweaks34 . We use the efficiency of our approach to record novel physical observables beyond the PTS correlation function that rely on efficiently sampling thermodynamic fluctuations inside the cavity. In particular, we follow the evolution with cavity size and temperature of the complete probability distribution function of the overlap. Its variance defines an overlap susceptibility that allows us to directly locate the PTS crossover scale without any empirical fitting. In addition, we are able to spatially resolve overlap fluctuations inside the cavity, giving us access both to their radial profile and orthoradial fluctuations, and thus making connection to PTS measurements
in other geometries, such as flat walls. The plan of this paper is as follows. In Sections II and III we detail the simulation model and the paralleltempering methodology, respectively. In Section IV we present the approach for computing PTS correlations as well as the results, and in Sec. V we evaluate results for the wall and penetration lengths from the cavity PTS setup. We briefly conclude in Section VI.
II.
SIMULATION MODEL
We study the behavior of the model glass-forming liquid first proposed by Kob and Andersen32,33 . The KobAndersen binary Lennard-Jones (KABLJ) model contains two particle species, denoted A and B, with equal mass m, and interacting via the pair potential σαβ 12 σαβ 6 Vαβ (r) = 4εαβ − , (1) r r where, α, β ∈ {A, B} with parameters εAB /εAA = 1.5, εBB /εAA = 0.5, σAB /σAA = 0.8, and σBB /σAA = 0.88. cut The interaction potential is cut off at rαβ = 2.5σαβ and shifted, such that the potential vanishes at the cutoff. The relative number of particles is NA : NB = 4 : 1 −3 and the overall number density is ρ = N/V = 1.2σAA . Note that we report below lengths and temperatures in standard dimensionless Lennard-Jones units set by σAA and εAA /kB . We use bulk samples with N = 135, 000 particles in a periodic cubic box, generated as described in Ref. 35, at temperatures T = 1.00, 0.80, 0.60, 0.51, 0.45. At each temperature, we take 50 equilibrated snapshots, separated by more than 2τα , where τα is the bulk structural relaxation time. Then, within each snapshot, we randomly select a position as cavity center. For each center, we fix the particles outside the cavities of radius R, and continue sampling configurations for the Ncav particles inside the cavity, as described in Sec. III. Note that in order to directly evaluate how the behavior of the cavity changes as function of R, we preserve each of the cavity centers as R increases.
III. PARALLEL-TEMPERING METHODOLOGY AND CONVERGENCE
A key hurdle to measuring PTS observables in simulations has been the large computational effort needed to sample different equilibrium configurations inside a cavity. Here, we design a Monte Carlo parallel-tempering scheme36 that sidesteps this difficulty by dramatically lowering barriers within the rugged free-energy landscape. The approach goes as follows. We prepare a = 1, ..., n replicas of a given configuration inside the cavity. The replicas evolve at different temperatures Ta and ‘shrinkage’ parameters λa ≤ 1 (see Appendix for
3 1
specific values) with a deformed Hamiltonian !12 !6 ˜ a σαβ ˜ a σαβ λ λ ˜ a ) = 4εαβ , (2) − Vαβ (r; λ r r
q¯(t) ≡
(t/trec ) X 1 q on (trec s) (t/trec ) s=1 c
(3)
decreases for the first scheme and increases for the second, converging upon equilibration (Fig. 1). The first seq configurations are discarded, and the overlap for the following sprod , hqcon i ≡
1 sprod
seq +sprod
X s=seq +1
qcon (trec s)
(4)
R = 3.2 0 1
R = 2.9 0 1
R = 2.6
q¯
˜ a = λa for a pair of mobile particles within a cavwhere λ ˜ a = 1+λa for a pair containing one mobile (inside ity and λ 2 the cavity) and one pinned particle (outside the cavity). We draw cavity configurations from the bottom replica with T1 = T and λ1 = 1, whereas the other replicas evolve at higher temperatures Ta > T and with smaller particles, λa < 1, in order to speed up their dynamics. Within each replica, we perform simple Monte Carlo (MC) moves that consist of: (i) choosing a particle i from Ncav mobile particles inside the cavity; (ii) displacing particle i by ∆x = lˆ n, where l is uniformly drawn from ˆ uniformly drawn on the sphere S 2 ; and [0, 0.3] and n (iii) accepting/rejecting the displacement according to the Metropolis criterion. Note that we put a hard spherical wall at the edge of the cavity, such that all moves that take a mobile particle outside the cavity radius are rejected. Each MC sweep consists of Ncav MC trial moves so that on average each particle attempts to move once. Most crucially, in order to release the disorder constraint (frustration) induced by the pinned particles, identityexchange of a pair of adjacent replicas [in (Ta , λa ) space] is attempted every 1000 MC sweeps, on average. This replica swapping is again accepted or rejected according to the Metropolis criterion, so that our MC algorithm ensures proper equilibrium sampling. Compared to earlier schemes for cavity studies, the proposed method is distinct from the simpler annealing procedure used before for the same model10 , and it is more generally applicable than the local particle swap Monte Carlo moves that are only efficient for specific glass-forming models8 . The quality of the equilibration within each cavity is evaluated by employing two schemes, following the approach developed by Cavagna et al.22 : we start the system from (i) the original configuration and (ii) a randomized configuration prepared by putting the cavity at T = 1.00 and λ = 0.6 for 104 MC sweeps. We then record the re-equilibrated configurations every trec = 104 MC sweeps and monitor the core overlap (as defined in Sect. IV) between the new configuration and the original cavity configuration, qcon (t), as a function of the number of MC sweeps, t. Their running average
R = 3.5 0 1
0 1
R = 2.3 0 1
R = 2.0 0 1
R = 1.7 0 1
R = 1.4 0
4
10
5
10
6
10
t
7
10
8
10
FIG. 1. Running average of core overlaps, q¯, after t MC sweeps from both the original (solid lines) and a randomized (dashed lines) configurations at T = 0.51 for a cavity of radius R. Convergence is faster in smaller cavities where parallel-tempering is more efficient.
is computed. Convergence is deemed obtained when the results for both approaches lie within ±qtol of each other for each cavity. Replica parameters as well as seq and sprod are chosen, such that for qtol = 0.1, at least 98% of cavities pass this convergence test (see Appendix for actual values). However, because the differences between the two approaches are not systematic, averaging over 50 cavities results in a very close agreement between the two schemes, namely a convergence to within ±0.005. This algorithm behaves differently in different cavity regimes. The most impressive results are obtained for small cavities, even at low temperatures. Only a few replicas are needed for the system to jump over the relatively high free-energy barriers and thus quickly equilibrate and properly sample configurations. This efficiency is illustrated in Fig. 1, which shows that the algorithm time to convergence is smaller for smaller cavities. This outcome markedly contrasts with the efficiency of simpler Monte Carlo scheme. (Note, however, that this computational efficiency depends sensitively on the details of the parallel-tempering parameters, which thus requires more fine-tuning than the parameters of simpler schemes.) As an example, results from Ref. 10 suggest that for low temperatures T ≤ 0.51 and small radii R ≤ 3.0, at least 1010 MC sweeps would be needed to equilibrate the system. Here, by contrast, we can equilibrate a similar system within ∼ 107 MC sweeps using ∼ 10 replicas. We thus conservatively achieve at least a 100-fold speedup, and the speedup would be even stronger for state points where conventional molecular dynamics does not converge. By contrast, for R ξPTS , confinement effects are negligible and even simple Monte Carlo moves suffice to sample configurations. The most problematic regime is R ∼ ξPTS at low temperatures where even relatively large cavities display large fine-tuning bottlenecks. Because the number
4 of replicas needed to√equilibrate a sample in this regime roughly grows like Ncav , it is unsurprising that this regime sets the lower limit on the convergence criterion. In earlier work22 , an acceleration of the overlap dynamics was reported in a binary soft sphere mixture, where traditional Monte Carlo moves are supplemented by binary particle exchanges (or ‘swap’ moves), a method which was used in a series of numerical work7,8,22 . We cannot directly compare our work to these dynamic measurements, as our parallel-tempering scheme does not allow us to measure time correlation functions, and no evolution of the convergence time was reported using particle swaps only22 . We believe, however, that the speedup of the convergence time that we report in Fig. 1 is of a different nature, and should uniquely be attributed to the merits of the parallel-tempering scheme and not to any natural dynamical process taking place in the glassforming liquid at various degrees of confinement. IV.
POINT-TO-SET OBSERVABLES
Once the glass-forming liquid is confined by amorphous boundaries in a finite cavity we need to analyse the equilibrium fluctuations of the overlap field q(r) inside the cavity, which is the principal observable used in this paper. To this end, we proceed as follows. Denote a pair of two configurations by X = {xi } and Y = {yi }. For each particle xi , find the nearest particle yinn of the same species, and assign an overlap value qX (xi ) ≡ w xi − yinn , where z 2 w(z) ≡ exp − , (5) b with b = 0.2. This function defines overlap values qX (xi ) at scattered points {xi }. We then define qX (r) to be a continuous function precisely passing through these points. Specifically, we first perform a Delaunay tessellation of space and, to a point r within a simplex spanned by four points {xi }i=i1 ,i2 ,i3 ,i4 , we associate a linearly interpolated value X qX (r) = ci qX (xi ) , (6) i=i1 ,i2 ,i3 ,i4
P where {ci }i=i1 ,i2 ,i3 ,i4 satisfies r = i=i1 ,i2 ,i3 ,i4 ci xi with P i=i1 ,i2 ,i3 ,i4 ci = 1. Similarly, we obtain qY (r), and define q (r) ≡
1 {qX (r) + qY (r)} , 2
(7)
which provides the overlap near the core of the cavity Z 3 dr q (r) , (8) qc ≡ 4πrc3 |r| 1) the linear relation λa − λ1 Ta − T1 = , Tdec − T1 λdec − λ1
(A.1)
where (Tdec , λdec ) is chosen such that the particles can decorrelate the overlap effectively. We always choose the last replica parameter λn ≥ λdec . The recording time is fixed as trec = 104 MC sweeps. As described in the main text, we discard the first seq configurations, keep the following sprod configurations, and compare the thermal averages [computed with Eq. (4)] obtained from two schemes: one starting from the original configuration and the other from a randomized configuration. The convergence criterion requires that results from both approaches lie within ±qtol of each other. For each data point, we record as success rate how many of 50 cavities satisfy this criterion for a particular qtol . Globally, for qtol = 0.1, 98% of cavities are deemed to have converged, for all temperatures and radii. Note that the parameters chosen are neither unique nor optimal. Some of the chosen parameters result in near bottlenecks in the exchanges of replicas. Tuning the replica parameters for each cavity by hand could certainly help sampling low-temperature configurations. A more promising and general way would be to algorithmically tune the replica parameters by monitoring upward and downward flows of replicas43 , in order to reduce the human time investment. R success seq sprod Tdec λdec {λa }
1.4 100% 1000 4000 1.0000 0.8000 1.0000 0.9500 0.9000 0.8500 0.8000
1.7 100% 1000 4000 1.0000 0.8000 1.0000 0.9667 0.9333 0.9000 0.8667 0.8333 0.8000
2.0 100% 1000 4000 1.0000 0.8000 1.0000 0.9750 0.9508 0.9271 0.9025 0.8773 0.8517 0.8260 0.7999
2.3 100% 1000 4000 1.0000 0.8000 1.0000 0.9800 0.9602 0.9404 0.9208 0.9015 0.8813 0.8608 0.8403 0.8200 0.7990
2.6 100% 1000 4000 1.0000 0.8000 1.0000 0.9835 0.9670 0.9507 0.9349 0.9190 0.9025 0.8862 0.8696 0.8524 0.8356 0.8178 0.8000
2.9 100% 1000 4000 1.0000 0.8000 1.0000 0.9870 0.9741 0.9610 0.9484 0.9357 0.9229 0.9103 0.8973 0.8842 0.8707 0.8569 0.8430 0.8288 0.8144 0.8000
3.5 100% 1000 4000 1.0000 1.0000 1.0000
TABLE I. Replica parameters for T = 1.00. Success rate is determined for qth = 0.025.
10 R success seq sprod Tdec λdec {λa }
1.4 98% 1000 4000 1.0000 0.8000 1.0000 0.9550 0.9100 0.8650 0.8200 0.7750
1.7 100% 1000 4000 1.0000 0.8000 1.0000 0.9700 0.9400 0.9120 0.8810 0.8493 0.8165 0.7830
2.0 98% 1000 4000 1.0000 0.8000 1.0000 0.9779 0.9556 0.9333 0.9111 0.8887 0.8649 0.8409 0.8167 0.7923
2.3 98% 1000 6000 1.0000 0.8000 1.0000 0.9812 0.9625 0.9444 0.9265 0.9086 0.8903 0.8723 0.8525 0.8332 0.8135 0.7935
2.6 96% 1000 4000 0.8000 0.8600 1.0000 0.9864 0.9730 0.9600 0.9475 0.9347 0.9208 0.9060 0.8909 0.8755 0.8600
2.9 100% 1000 4000 0.8000 0.8800 1.0000 0.9894 0.9789 0.9686 0.9585 0.9484 0.9383 0.9283 0.9174 0.9054 0.8930 0.8800
4.0 100% 1000 4000 0.8000 1.0000 1.0000
TABLE II. Replica parameters for T = 0.80. Success rate is determined for qth = 0.03.
R success seq sprod Tdec λdec {λa }
1.4 100% 1000 4000 1.0000 0.8000 1.0000 0.9600 0.9200 0.8800 0.8400 0.7990
1.7 100% 1000 4000 1.0000 0.8000 1.0000 0.9725 0.9440 0.9161 0.8888 0.8600 0.8292 0.7968
2.0 100% 1000 4000 1.0000 0.8000 1.0000 0.9805 0.9600 0.9400 0.9201 0.9000 0.8790 0.8567 0.8330 0.8086 0.7830
2.3 98% 1000 7000 0.8000 0.8200 1.0000 0.9859 0.9721 0.9579 0.9445 0.9313 0.9182 0.9041 0.8893 0.8731 0.8560 0.8382 0.8200
2.6 100% 1000 9000 0.8000 0.8600 1.0000 0.9877 0.9751 0.9627 0.9507 0.9389 0.9272 0.9153 0.9022 0.8888 0.8748 0.8600
2.9 98% 2000 8000 0.8000 0.8800 1.0000 0.9904 0.9807 0.9711 0.9619 0.9528 0.9435 0.9340 0.9239 0.9136 0.9028 0.8915 0.8800
3.2 98% 2000 13000 0.8000 0.8900 1.0000 0.9919 0.9839 0.9762 0.9685 0.9611 0.9536 0.9459 0.9373 0.9283 0.9191 0.9096 0.8998 0.8900
TABLE III. Replica parameters for T = 0.60. Success rate is determined for qth = 0.06.
5.0 100% 1000 4000 0.6000 1.0000 1.0000
11 R success seq sprod Tdec λdec {λa }
1.4 100% 1000 4000 1.0000 0.8000 1.0000 0.9600 0.9200 0.8800 0.8400 0.7970
1.7 100% 1000 4000 1.0000 0.8000 1.0000 0.9760 0.9500 0.9240 0.8990 0.8735 0.8466 0.8178 0.7870
2.0 100% 1000 4000 1.0000 0.8000 1.0000 0.9825 0.9640 0.9450 0.9250 0.9050 0.8850 0.8640 0.8423 0.8200 0.7960
2.3 100% 1000 7000 0.8000 0.8200 1.0000 0.9873 0.9744 0.9616 0.9491 0.9372 0.9257 0.9137 0.9011 0.8874 0.8724 0.8547 0.8378 0.8200
2.6 100% 1000 9000 0.8000 0.8600 1.0000 0.9890 0.9779 0.9668 0.9559 0.9454 0.9351 0.9243 0.9127 0.9005 0.8877 0.8741 0.8600
2.9 100% 3000 7000 0.8000 0.8800 1.0000 0.9916 0.9832 0.9747 0.9666 0.9582 0.9500 0.9415 0.9325 0.9232 0.9132 0.9026 0.8915 0.8800
3.2 98% 3000 12000 0.6000 0.8900 1.0000 0.9946 0.9892 0.9839 0.9787 0.9737 0.9688 0.9638 0.9590 0.9544 0.9499 0.9423 0.9343 0.9258 0.9171 0.9083 0.8992 0.8900
TABLE IV. Replica parameters for T = 0.51. Success rate is determined for qth = 0.09.
R success seq sprod Tdec λdec {λa }
1.4 100% 1000 4000 1.0000 0.8000 1.0000 0.9667 0.9333 0.8950 0.8570 0.8170 0.7710
1.7 100% 1000 4000 1.0000 0.8000 1.0000 0.9760 0.9520 0.9280 0.9040 0.8795 0.8548 0.8263 0.7960
2.0 100% 1000 4000 1.0000 0.8000 1.0000 0.9830 0.9650 0.9460 0.9270 0.9070 0.8890 0.8680 0.8465 0.8235 0.8000
2.3 100% 1000 4000 0.8000 0.8200 1.0000 0.9879 0.9753 0.9627 0.9504 0.9387 0.9273 0.9159 0.9035 0.8882 0.8729 0.8560 0.8381 0.8200
2.6 100% 2000 6000 0.8000 0.8600 1.0000 0.9913 0.9824 0.9735 0.9645 0.9560 0.9478 0.9396 0.9312 0.9224 0.9124 0.8997 0.8866 0.8736 0.8600
2.9 100% 2000 10000 0.6000 0.8800 1.0000 0.9937 0.9876 0.9816 0.9755 0.9693 0.9632 0.9574 0.9514 0.9458 0.9397 0.9305 0.9209 0.9109 0.9006 0.8905 0.8800
3.2 98% 3000 17000 0.6000 0.8900 1.0000 0.9949 0.9898 0.9847 0.9799 0.9749 0.9699 0.9652 0.9605 0.9557 0.9506 0.9427 0.9345 0.9259 0.9171 0.9084 0.8992 0.8900
TABLE V. Replica parameters for T = 0.45. Success rate determined for qth = 0.10.
3.5 98% 4000 16000 0.6000 0.9000 1.0000 0.9953 0.9906 0.9861 0.9817 0.9772 0.9729 0.9688 0.9646 0.9605 0.9565 0.9526 0.9458 0.9386 0.9311 0.9235 0.9159 0.9080 0.9000
6.0 100% 1000 4000 0.5100 1.0000 1.0000
